Infinite-dimensional Space Research Articles

A structured L-BFGS method and its application to inverse problems

Many inverse problems are phrased as optimization problems in which the objective function is the sum of a data-fidelity term and a regularization. Often, the Hessian of the fidelity term is computationally unavailable while the Hessian of the regularizer allows for cheap matrix-vector products. In this paper, we study an L-BFGS method that takes advantage of this structure. We show that the method converges globally without convexity assumptions and that the convergence is linear under a Kurdyka–Łojasiewicz-type inequality. In addition, we prove linear convergence to cluster points near which the objective function is strongly convex. To the best of our knowledge, this is the first time that linear convergence of an L-BFGS method is established in a non-convex setting. The convergence analysis is carried out in infinite dimensional Hilbert space, which is appropriate for inverse problems but has not been done before. Numerical results show that the new method outperforms other structured L-BFGS methods and classical L-BFGS on non-convex real-life problems from medical image registration. It also compares favorably with classical L-BFGS on ill-conditioned quadratic model problems. An implementation of the method is freely available.

Evaluation of data-driven neural operators in ocean acoustic propagation modeling

Various types of neural networks have been employed to tackle a wide range of complex partial differential equations (PDEs) and ordinary differential equations (ODEs). Notably, neural operators such as DeepONet and FNO show promise in handling these problems, offering potential real-time prediction capabilities. In contrast to physics-informed neural network (PINN) methods, neural operators primarily derive insight from extensive, well-prepared datasets. In the context of ocean acoustic propagation modeling, where the challenge involves solving the wave or Helmholtz equation given specific boundary conditions, this study focuses on assessing the performance of data-driven neural operators in predicting sound pressure. Unlike conventional approaches that map between finite-dimensional Euclidean spaces, neural operators excel in learning mappings between infinite-dimensional function spaces—a particularly advantageous feature in sound propagation modeling tasks. This research specifically delves into evaluating the generalization capabilities of neural operators when applied to sound propagation modeling in a range-independent shallow water environment. By exploring the neural operators' effectiveness in this domain, the study aims to contribute valuable insights into their potential applications for real-world ocean acoustics simulations.

함수형 시공간 기법을 활용한 지표면 온도의 변화점 분석

Weather stations collect various meteorological variables such as temperature, precipitation, and humidity. These variables have been measured at the regular and repetitive time points at the specific location in the year. However, ideally, the characteristic of the climate data is that these variables exist in the form of a continuous function regardless of the time points observed. This characteristic is considered to be close to the functional data perspective, which assumes that data are observed in the form of function values at discrete points from a random function that exists as a continuous function in an infinite-dimensional space, and they require statistical techniques that are different from existing approaches. In this study, we aim to analyze land surface temperature data to determine that there is a change point in the spatial average for almost domestic observatories, assuming that they measured over a total of 42 years from 1981 to 2022 at 48 sites is functional data. Specifically, after estimating the spatial and temporal covariances, the weight for integrating spatial information was extracted from the spatial covariance structure, and the eigenfunctions and eigenvalues for integrating temporal information were extracted from the temporal covariance structure. Based on this, we introduce a method of expressing statistics as a partial sum of the mean difference before and after a specific point in time, as well as an approximate distribution.

The Existence of Solutions for w-Weighted ψ-Hilfer Fractional Differential Inclusions of Order μ ∈ (1, 2) with Non-Instantaneous Impulses in Banach Spaces

In this research, we obtain the sufficient conditions that guarantee that the set of solutions for an impulsive fractional differential inclusion involving a w-weighted ψ-Hilfer fractional derivative, D0,tσ,v,ψ,w,of order μ∈(1,2), in infinite dimensional Banach spaces that are not empty and compact. We demonstrate the exact relation between a differential equation involving D0,tσ,v,ψ,w of order μ ∈(1,2) in the presence of non-instantaneous impulses and its corresponding fractional integral equation. Then, we derive the formula for the solution for the considered problem. The desired results are achieved using the properties of the w-weighted ψ-Hilfer fractional derivative and appropriate fixed-point theorems for multivalued functions. Since the operator D0,tσ,v,ψ,w includes many types of well-known fractional differential operators, our results generalize several results recently published in the literature. We give an example that illustrates and supports our theoretical results.

Existence of Walrasian equilibria with discontinuous, non-ordered, interdependent preferences, without free disposal, and with an infinite-dimensional commodity space

A new proof of the existence of a Walrasian equilibrium with an infinite dimensional commodity space is provided, which allows agents’ preferences to be discontinuous. The new theorems include as corollaries the existence results of Mas-Collel, Yannelis and Zame, Araujo and Monteiro, and Mas-Collel and Richard, among others.

On the approximation of bi-Lipschitz maps by invertible neural networks

Invertible neural networks (INNs) represent an important class of deep neural network architectures that have been widely used in applications. The universal approximation properties of INNs have been established recently. However, the approximation rate of INNs is largely missing. In this work, we provide an analysis of the capacity of a class of coupling-based INNs to approximate bi-Lipschitz continuous mappings on a compact domain, and the result shows that it can well approximate both forward and inverse maps simultaneously. Furthermore, we develop an approach for approximating bi-Lipschitz maps on infinite-dimensional spaces that simultaneously approximate the forward and inverse maps, by combining model reduction with principal component analysis and INNs for approximating the reduced map, and we analyze the overall approximation error of the approach. Preliminary numerical results show the feasibility of the approach for approximating the solution operator for parameterized second-order elliptic problems.

A Fourier neural operator-based lightweight machine learning framework for topology optimization

The combination of machine learning and topology optimization has preliminarily mitigated the enormous computational costs incurred during optimization processes. Its powerful design capabilities are warmly welcomed by many scholars, but this strategy still faces many problems, such as unclear structural boundaries and high dependence on big data. Here, we propose a new machine learning topology optimization paradigm that learns the mapping relationship between the initial and optimized Level set functions in the infinite-dimensional function space under the Fourier neural operator to truly parameterize the structural model and its mechanical information, thereby achieving purely data-driven topology optimization. A base model trained under the proposed approach could obtain a topological structure with clear and smooth structural boundaries within 0.4 seconds, making it faster than traditional methods by tens of times. In addition, a lightweight retraining strategy under the base model was proposed. Four additional numerical examples were used to demonstrate that this strategy requires only 100 or 200 new samples and 3-6 seconds of retraining time to attain prediction capabilities under different boundary conditions, different meshes (resolution), various initial level set functions, and multiple loads. In addition, we discussed all the experiments and found that most of the mechanical properties of the results were close to the ground truths, while some results were even better than those of the traditional method. Although we only used the maximum structural stiffness as an example, this method can be extended to different objective functions.

A note on a generalized Jordan form of an infinite upper triangular matrix

In this paper, several equivalent conditions for the existence of a generalized Jordan form for matrices of locally nilpotent linear operators acting on an infinite countable dimensional vector space are presented, and an example is given showing that these conditions are not always fulfilled, which means the existence of infinite upper triangular matrices that are not similar to any generalized infinite Jordan matrix. Besides, it is shown that these conditions are not applicable in the case of an arbitrary dimensional of a vector space.

Quasitopological vector spaces

In this paper, motivated by a diffeological vector space with the D-topology, we introduce the concept of quasitopological vector space, which is a vector space with a topology satisfying the conditions that the vector addition is separately continuous in each variable, and the scalar multiplication is jointly continuous. The following results are proved: (1) Every quasitopological finite-dimensional vector space is a topological vector space. (2) For each infinite-dimensional vector space, we define a T1 non-Hausdorff topology on it such that it is a quasitopological vector space but is not a topological vector space.

Polynomial convergence of iterations of certain random operators in Hilbert space

Abstract We study the convergence of a random iterative sequence of a family of operators on infinite-dimensional Hilbert spaces, inspired by the stochastic gradient descent (SGD) algorithm in the case of the noiseless regression. We identify conditions that are strictly broader than previously known for polynomial convergence rate in various norms, and characterize the roles the randomness plays in determining the best multiplicative constants. Additionally, we prove almost sure convergence of the sequence.

On linear non-uniform cellular automata: Duality and dynamics

For linear non-uniform cellular automata (NUCA) which are global perturbations of CA over an arbitrary universe, we introduce and investigate their dual linear NUCA, which are also endomorphisms over a generally infinite dimensional vector space. Generalizing results for linear CA, we show that dynamical properties namely pre-injectivity, resp. injectivity, resp. stably injectivity, resp. invertibility of a linear NUCA is equivalent to surjectivity, resp. post-surjectivity, resp. stably post-surjectivity, resp. invertibility of the dual linear NUCA. However, while bijectivity is a dual property for linear CA, it is no longer the case for linear NUCA. We prove that for linear NUCA, stable injectivity and stable post-surjectivity are precisely characterized respectively by left invertibility and right invertibility and that a linear NUCA is invertible if and only if it is pre-injective and stably post-surjective. Surprisingly, we show that linear NUCA with finite memory satisfy the shadowing property also known as the pseudo-orbit tracing property. Thus, linear NUCA can be useful for fault-tolerant computation. Applications on the dual surjunctivity are also obtained.

Lipschitz continuity of the dilation of Bloch functions on the unit ball of a Hilbert space and applications

Let BE\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_E$$\\end{document} be the open unit ball of a complex finite- or infinite-dimensional Hilbert space. If f belongs to the space B(BE)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {B}(B_E)$$\\end{document} of Bloch functions on BE\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_E$$\\end{document}, we prove that the dilation map given by x↦(1-‖x‖2)Rf(x)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x \\mapsto (1-\\Vert x\\Vert ^2) \\mathcal {R}f(x)$$\\end{document} for x∈BE\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x \\in B_E$$\\end{document}, where Rf\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {R}f$$\\end{document} denotes the radial derivative of f, is Lipschitz continuous with respect to the pseudohyperbolic distance ρE\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rho _E$$\\end{document} in BE\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_E$$\\end{document}, which extends to the finite- and infinite-dimensional setting the result given for the classical Bloch space B\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {B}$$\\end{document}. To provide this result, we will need to prove that ρE(zx,zy)≤|z|ρE(x,y)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rho _E(zx,zy) \\le |z| \\rho _E(x,y)$$\\end{document} for x,y∈BE\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x,y \\in B_E$$\\end{document} under some conditions on z∈C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$z \\in \\mathbb {C}$$\\end{document}. Lipschitz continuity of x↦(1-‖x‖2)Rf(x)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x \\mapsto (1-\\Vert x\\Vert ^2) \\mathcal {R}f(x)$$\\end{document} will yield some applications on interpolating sequences for B(BE)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {B}(B_E)$$\\end{document} which also extends classical results from B\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {B}$$\\end{document} to B(BE)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {B}(B_E)$$\\end{document}. Indeed, we show that it is necessary for a sequence in BE\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_E$$\\end{document} to be separated to be interpolating for B(BE)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {B}(B_E)$$\\end{document} and we also prove that any interpolating sequence for B(BE)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {B}(B_E)$$\\end{document} can be slightly perturbed and it remains interpolating.

Continuous-Variable Quantum State Designs: Theory and Applications

We generalize the notion of quantum state designs to infinite-dimensional spaces. We first prove that, under the definition of continuous-variable (CV) state t-designs from [Blume-Kohout , ], no state designs exist for t≥2. Similarly, we prove that no CV unitary t-designs exist for t≥2. We propose an alternative definition for CV state designs, which we call rigged t-designs, and provide explicit constructions for t=2. As an application of rigged designs, we develop a design-based shadow-tomography protocol for CV states. Using energy-constrained versions of rigged designs, we define an average fidelity for CV quantum channels and relate this fidelity to the CV entanglement fidelity. As an additional result of independent interest, we establish a connection between torus 2-designs and complete sets of mutually unbiased bases. Published by the American Physical Society 2024

Chow's theorem for Hilbert Grassmannians as a Wigner-type theorem

Let H be an infinite-dimensional complex Hilbert space. Denote by G ∞ ( H ) the Grassmannian formed by closed subspaces of H whose dimension and codimension both are infinite. We say that X , Y ∈ G ∞ ( H ) are ortho-adjacent if they are compatible and X ∩ Y is a hyperplane in both X, Y. A subset C ⊂ G ∞ ( H ) is called an A-component if for any X , Y ∈ C the intersection X ∩ Y is of the same finite codimension in both X, Y and C is maximal with respect to this property. Let f be a bijective transformation of G ∞ ( H ) preserving the ortho-adjacency relation in both directions. We show that the restriction of f to every A-component is induced by a unitary or anti-unitary operator or it is the composition of the orthocomplementary map and a map induced by a unitary or anti-unitary operator. Note that the restrictions of f to distinct A-components can be related to different operators.

Dynamically Meaningful Latent Representations of Dynamical Systems

Dynamical systems are ubiquitous in the physical world and are often well-described by partial differential equations (PDEs). Despite their formally infinite-dimensional solution space, a number of systems have long time dynamics that live on a low-dimensional manifold. However, current methods to probe the long time dynamics require prerequisite knowledge about the underlying dynamics of the system. In this study, we present a data-driven hybrid modeling approach to help tackle this problem by combining numerically derived representations and latent representations obtained from an autoencoder. We validate our latent representations and show they are dynamically interpretable, capturing the dynamical characteristics of qualitatively distinct solution types. Furthermore, we probe the topological preservation of the latent representation with respect to the raw dynamical data using methods from persistent homology. Finally, we show that our framework is generalizable, having been successfully applied to both integrable and non-integrable systems that capture a rich and diverse array of solution types. Our method does not require any prior dynamical knowledge of the system and can be used to discover the intrinsic dynamical behavior in a purely data-driven way.

Hasimoto frames and the Gibbs measure of the periodic nonlinear Schrödinger equation

The paper interprets the cubic nonlinear Schrödinger equation as a Hamiltonian system with infinite dimensional phase space. There exists a Gibbs measure which is invariant under the flow associated with the canonical equations of motion. The logarithmic Sobolev and concentration of measure inequalities hold for the Gibbs measures, and here are extended to the k-point correlation function and distributions of related empirical measures. By Hasimoto’s theorem, the nonlinear Schrödinger equation gives a Lax pair of coupled ordinary differential equations for which the solutions give a system of moving frames. The paper studies the evolution of the measure induced on the moving frames by the Gibbs measure; the results are illustrated by numerical simulations. The paper contains quantitative estimates with well-controlled constants on the rate of convergence of the empirical distribution in Wasserstein metric.

Beyond the finite: An exploration of infinite-dimensional vector spaces

In this paper, we delve deeply into the intricacies of linear algebra, with a focus on the progression from finite to infinite-dimensional vector spaces. Starting with the foundational concepts, we define vectors, vector spaces, linear combinations, and basis. The importance of infinite-dimensional vector spaces is emphasized, particularly their role in better understanding and modeling complex mathematical phenomena. Through well-illustrated examples, we guide the reader on how to validate if a given set can be classified as a vector space. Additionally, the methodology to identify bases for these vast spaces is discussed in detail. Reduction methods also play an important role in determining bases for infinite-dimensional spaces. In our conclusion, we reflect on the evolution from basic vector concepts to the more nuanced understanding of infinite dimensions. This progression not only deepens our understanding of vectors but also sets the stage for advanced investigations into linear relationships and transformations. By bridging the gap between elementary vector knowledge and advanced infinite-dimensional spaces, this paper makes a notable contribution to the ever-evolving field of linear algebra, serving as a valuable resource for both students and practitioners.

Infinitesimal gradient boosting

We define infinitesimal gradient boosting as a limit of the popular tree-based gradient boosting algorithm from machine learning. The limit is considered in the vanishing-learning-rate asymptotic, that is when the learning rate tends to zero and the number of gradient trees is rescaled accordingly. For this purpose, we introduce a new class of randomized regression trees bridging totally randomized trees and Extra Trees and using a softmax distribution for binary splitting. Our main result is the convergence of the associated stochastic algorithm and the characterization of the limiting procedure as the unique solution of a nonlinear ordinary differential equation in a infinite dimensional function space. Infinitesimal gradient boosting defines a smooth path in the space of continuous functions along which the training error decreases, the residuals remain centered and the total variation is well controlled.

Matrix representations of arbitrary bounded operators on Hilbert spaces

Abstract We show that under natural and quite general assumptions, a large part of a matrix for a bounded linear operator on a Hilbert space can be preassigned. The result is obtained in a more general setting of operator tuples leading to interesting consequences, e.g., when the tuple consists of powers of a single operator. We also prove several variants of this result of independent interest. The paper substantially extends former research on matrix representations in infinite-dimensional spaces dealing mainly with prescribing the main diagonals.

Darboux-Kadomtsev-Petviashvili system as an integrable Chern-Simons multiform theory in infinite dimensional space

The Darboux-Kadomtsev-Petviashvili system is a universal three-dimensional integrable field theory, which is a generating system for the entire Kadomtsev-Petviashvili hierarchy, and at the same time generalizes the Darboux system, describing orthogonal curvilinear coordinates. In this paper, we establish a hierarchy of Lagrangian multiforms for the Darboux-Kadomtsev-Petviashvili system, derived from a hierarchy of Chern-Simons actions in an infinite-dimensional space of Miwa variables. This provides an integrable variational description of this multidimensionally consistent field theory embedded in infinite dimensional space. Published by the American Physical Society 2024